1Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, School of Optoelectronic Science and Engineering, South China Normal University, Guangzhou 510631, China
We propose an approach for generating the enhanced superchiral needle by matching electromagnetic components of the combined field, which is the superposition of a radially polarized vortex Bessel–Gaussian beam (RPVBGB) and an azimuthally polarized Bessel–Gaussian beam (APBGB). In the tightly focused combined field, the longitudinal magnetic component provided by the APBGB, together with the longitudinal electric component provided by the RPVBGB, induces an additional contribution to the optical chirality and thereby significantly improves the enhancement factor of the superchiral needle. It is revealed that the characteristics of the superchiral needle are mainly influenced by the ring aperture, the phase difference, and the amplitude ratio. Under proper parameters, the enhancement factor can reach from 22.9 to 32.9, and the needle width can reach from to and from to in the - and -directions, respectively. The results would be of interest for the chirality measurement of individual molecules.
【AIGC One Sentence Reading】:Enhanced superchiral needle generated by matching EM components of combined field, achieving high enhancement factor and narrow width.
【AIGC Short Abstract】:We propose enhancing superchirality by matching EM components of a combined field, achieved through superposition of radially and azimuthally polarized Bessel-Gaussian beams. This combination significantly boosts the enhancement factor and improves chirality, influenced by ring aperture, phase difference, and amplitude ratio. Results show potential for chirality measurement of single molecules.
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1. INTRODUCTION
When illuminated with left or right circularly polarized light (CPL), chiral molecules exhibit different absorption rates, an effect known as circular dichroism (CD) [1–5]. This difference can be represented with the dissymmetry factor , where and are the absorption rates under the excitation of left and right CPL, respectively. Conventional CD measurements are usually used for measuring samples over large areas. However, to distinguish chirality information of individual molecules, the response signal of conventional CD measurements is so weak that the chirality of the field is eagerly expected to be enhanced. In recent years, various methods for enhancing the chirality have been proposed, such as using nanostructures [6–14] or superchiral fields [15–17], which we focus on in this paper.
The superchiral field is a field whose dissymmetry factor is larger than that of the CPL. There are two formulas to describe the enhancement of superchirality. For most molecules, the contribution of the magnetic polarizability is much smaller than that of the electric polarizability so that it can be neglected. In this case, Tang et al. demonstrated that the enhancement factor of the superchiral field can be expressed as [15]
Here, is the wave impedance in the vacuum. Obviously, the superchiral field requires that [15,18–22].
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However, when the magnetic field is significantly enhanced compared to the electric field (such as at the nodes of superchiral standing waves [22]) so that the contribution of the magnetic polarizability cannot be neglected, the full expression containing the magnetic polarizability should be reformulated as [22] where is a factor determined by the material chosen, the range of which is typically [4].
In most cases, the contribution of the magnetic polarizability is negligibly small. Therefore, many previous works represent the superchirality enhancement with Eq. (1) [17,18,23,24], based on which the superchirality condition can be easily obtained.
Recently, some methods have been proposed to improve the enhancement factor . Tang et al. proposed the generation of superchiral fields at the nodes of a CPL standing wave [15,18]. The result shows that the enhancement factor is for the enantiomer and for the enantiomer. Li et al. proposed a tightly focused system capable of generating multiple spots with controllable enhanced chirality [25]. On the basis of the tightly focused system, Zhang et al. [24] introduced a superchiral field with and revealed a novel chiral mechanical effect. Based on the radially polarized vortex beam focused by a high numerical aperture (NA) lens, Hu et al. proposed a method for generating a superchiral needle (a needle-like distribution of the enhancement factor ) with the enhancement factor and the full-width at half-maximum (FWHM) being [17]. Then Rui et al. improved Hu’s method by focusing a radially polarized vortex beam onto a one-dimensional photonic bandgap structure and obtaining a needle with with the FWHM being [23].
In this paper, we aim to provide an approach for achieving the enhanced superchirality of the tightly focused combined field through matching the electromagnetic components of the constituent beams. The combined field is structured by the superposition of a radially polarized vortex Bessel–Gaussian beam (RPVBGB) containing a phase vortex with the topological charge of 1 and an azimuthally polarized Bessel–Gaussian beam (APBGB) without a phase vortex. The longitudinal magnetic component provided by the APBGB, together with the longitudinal electric component provided by the RPVBGB, yields a significant improvement of the optical chirality and thereby produces an enhanced superchiral needle. The influence of the ring aperture, the phase difference, and the amplitude ratio on the characteristics of the superchiral needle is also investigated.
2. THEORETICAL MODEL
We construct the enhanced superchiral needle by tightly focusing a combined field resulting from the superposition of an RPVBGB and an APBGB. As sketched in Fig. 1, a linearly polarized Bessel–Gaussian laser beam passes through a half-wave plate (HP) and a polarizing beam splitter (PBS), which divides the beam into two orthogonally polarized beams. The two beams are converted into the radially and azimuthally polarized beams by the S-wave plates (SP), respectively. The radially polarized beam is transformed into the RPVBGB by adding a vortex phase with the topological charge of 1 from the spiral phase plate (SPP). A thin plate (TP) is located in one branch to adjust the phase difference. The two beams are combined by a non-polarizing beam splitter (BS). The combined beam is limited by a ring aperture (RA), which is used to filter out the lower spatial frequency components and retain the higher spatial frequency components close to the critical angle for total reflection at the subsequent glass–air interface. Then it is tightly focused by a high NA oil-immersion objective (Obj), whose focal plane is at the glass–air interface. The focused incident field undergoes reflection and transmission at the interface. The transmitted field becomes the enhanced superchiral field.
Figure 1.Schematic diagram of the designed system for generating the enhanced superchiral needle. HP, half-wave plate; PBS, polarizing beam splitter; SP, S-wave plate; SPP, spiral phase plate; BS, beam splitter; TP, thin plate; RA, ring aperture; Obj, objective.
Considering the sine condition, the electric field of the combined beam before focusing can be written as where is the incident angle at the interface, , is the beam waist, , is the focal length of the lens, is the refractive index of the glass and oil (the oil and cover glass are assumed to have the same refractive index for simplifying the analysis), is the th order Bessel function, the term means that there is a phase vortex with the topological charge of 1 in the RPVBGB, is the azimuthal angle, is the phase difference adjusted by the thin plate, and are the unit vectors in the radial and azimuthal directions, respectively, and is twice the angle between the light polarization direction and the fast axis of the half-wave plate. The ratio coefficient of the two constituent beams is (APBGB:RPVBGB), which can be controlled by the half-wave plate.
We use the Richards–Wolf method to calculate the electromagnetic field distribution near the focus under tight focusing [26–30]. In this framework, the focused incident electromagnetic field at the glass–air interface can be expressed as where , , “” and “” represent electromagnetic fields coming from the RPVBGB (containing a phase vortex with the topological charge of 1) and the APBGB (without a phase vortex), respectively, and is the impedance of the optical waves in the glass or oil. The upper limit of integration is set as the critical angle of total reflection , in that the field would be transformed into evanescent waves with no chirality if the incident angle is larger than . The parameter () is determined by the slit width of the ring aperture.
Now, let us calculate the transmitted electromagnetic field at the glass–air interface, which can be obtained by decomposing the incident field into plane waves. The RPVBGB and APBGB fields could be decomposed into p-polarized and s-polarized plane waves, respectively. Based on Fresnel’s law, the transmission coefficients and for p-polarized and s-polarized plane waves can be written as where is the refracting angle, and is the refractive index of the air. Thus, the transmitted field is obtained as where and is the impedance of optical waves in the air.
For the convenience of comparison with Ref. [17] (RPVBGB used alone), we will first use [Eq. (1)] to investigate the superchirality enhancement of the combined field with a relatively small amplitude ratio () as a representative case, for which Eq. (1) is applicable. Then, the modified enhancement factor [Eq. (2)] will be used in the case of the large amplitude ratio.
Based on Eqs. (1) and (11), one can get the enhancement factor of the superchiral field where is the optical chirality induced by the corresponding electromagnetic component, and .
3. RESULTS AND DISCUSSION
As shown in Fig. 2, the combined field significantly improves the enhancement factor compared to the RPVBGB used alone in Ref. [17]. When , the maximum value of reaches 23.4, which is almost twice as that in Ref. [17] [Fig. 2(c)]. In addition, The needle width is nearly half that in Ref. [17].
Figure 2.(a) Distribution of the enhancement factor at the focal plane. (b) Distribution of the enhancement factor at the plane (). (c) Lateral distribution in the -direction of the enhancement factor at the focal plane. , , , , , , and .
The key to enhancing superchirality in our system lies in achieving the overlap of a strong magnetic component and a weak electric component within a small region by matching the electromagnetic components of the combined field (since the superchiral condition is ). A single tightly focused RPVBGB can produce a doughnut-shaped total electric field and a bell-shaped total magnetic field at the beam center [Figs. 3(n) and 3(p)], creating a small region with the strong magnetic and weak electric fields that enables the generation of the superchiral needle [17]. However, this configuration lacks the longitudinal magnetic component, i.e., [Fig. 3(l)], resulting in no longitudinal component in the superchiral needle, i.e., [Fig. 4(f)]. To provide the longitudinal superchirality component, in our system we additionally introduce a non-vortex APBGB to generate a strong bell-shaped longitudinal magnetic component at the beam center [Fig. 3(k)]. This bell-shaped longitudinal magnetic component, together with the RPVBGB’s doughnut-shaped longitudinal electric component [Fig. 3(j)], yields a strong longitudinal superchirality component in a small region [Fig. 4(c)], which significantly improves the total enhancement factor and thereby generates the enhanced superchiral needle.
Figure 3.Comparison of the distributions for the electromagnetic components at the focal plane between the combined field (columns 1 and 3, ) and the RPVBGB alone (columns 2 and 4, ). The parameters are the same as Fig. 2.
Figure 4.Comparison of the distributions for , , and at the focal plane between the combined field (upper row, ) and the RPVBGB alone (bottom row, ). The parameters are the same as Fig. 2.
It should be noted that, although the APBGB in the combined field plays a key role for the enhancement of superchirality, a single APBGB alone (corresponding to ) exhibits no chirality because in this case , , (yielding ).
Although both the APBGB and the RPVBGB exhibit axisymmetric intensity and polarization distributions, the APBGB lacks a phase vortex while the RPVBGB carries a phase vortex. Therefore, the phase difference between the two beams varies at different azimuthal angles, leading to a non-axisymmetric distribution of the combined field (induced by coherent superposition of the two beams), which in turn produces a non-axisymmetric distribution of the enhancement factor [Fig. 2(a)].
The ring aperture plays a key role in producing the distribution of the enhancement factor . Since it suppresses lower spatial frequency components, a narrower slit of the ring aperture results in stronger longitudinal components and thereby induces a stronger longitudinal enhancement factor component , which makes the main contribution to the total enhancement factor . Therefore, with the decrease of the slit width of the ring aperture (i.e., the increase of ), the maximum of for the superchiral needle in the combined field monotonically increases [Fig. 5(a)]. In addition, As shown in Fig. 5(b), the FWHM for the superchiral needle decreases with the decrease of slit width of the ring aperture. Now that the maximum of (the FWHM) for the superchiral needle in the combined field remains much larger (smaller) than that of the pure RPVBGB, it makes the chirality measurement experiment more flexible. For example, to obtain the same maximum of the enhancement factor (needle width) of the combined field as that of the pure RPVBGB, a larger slit width could be allowed. Moreover, the slit width of the ring aperture also affects the location of the superchiral needle. Specifically, the narrower the slit is, the closer the superchiral needle of the combined field is to the -axis [Fig. 5(c)].
Figure 5.Comparison of the parameters for the superchiral needle at the focal plane for various values of between the combined field () and the RPVBGB alone (). (a) The maximum of the enhancement factor . (b) The FWHM in the -direction. (c) The deviation of the superchiral needle from the -axis (). The parameters are the same as Fig. 2.
As shown in Fig. 6, the phase difference between the two beams does not influence the magnitude of the superchirality. However, the needle undergoes anticlockwise rotation with the increase of the phase difference , and the variation of its azimuthal orientation is equal to the variation of [Fig. 6(e)]. The physical explanation for this effect is as follows: the phase vortex in the RPVBGB makes the phase difference between the two constituent beams vary at different azimuthal angles. Thus, for a fixed total phase difference (i.e., the vortex phase minus the phase difference induced by the thin plate), the azimuthal angle increases synchronously with . This results in the distribution of the combined field rotating anticlockwise by an angle equal to , thereby inducing a corresponding rotation of the distribution with the same angle. Combining this property and the fact that the location can be steered by the slit width, we can manipulate the location of the superchiral optical needle in the chiral measurement.
Figure 6.(a)–(d) Distribution of the enhancement factor at the focal plane for different phase differences . The parameters are the same as Fig. 2. (e) The maximum of as a function of (blue line, left ordinate). The azimuthal orientation () of the distribution for as a function of (green line, right ordinate).
The amplitude ratio is an important parameter for adjusting the strength of the superchirality (Fig. 7). When we adjust the half-wave plate such that approaches 0°, the focused APBGB field becomes negligible, and the value of the enhancement factor is very close to that when the RPVBGB is used alone [17]. In the case of , the enhancement factor is strictly equal to 0 because the APBGB field alone does not exhibit optical chirality.
Figure 7.(a) Lateral maximum of the enhancement factor [represented as ] as a function of . Solid line: resulting from the simplified model [Eq. (1)]. Dashed line: resulting from the reformulated model [Eq. (2)]. (b), (c) The lateral distribution in the - (b) and - (c) directions of the modified enhancement factor when reaches the maximum in (a) for different values of . The parameters are the same as Fig. 2.
However, it should be noted that, when approaches 90°, the maximum of approaches a nonzero value, resulting in a mathematical discontinuity at [Fig. 7(a), solid purple line]. This obviously does not conform to the physical expectation, which suggests that the model should be reformulated [22].
In fact, when the amplitude ratio becomes too large, compared with the RPVBGB alone, the magnetic field of the combined beam will be significantly enhanced compared to the electric field. Under this condition, the interaction between the magnetic field and the matter cannot be ignored, and the modified enhancement factor [i.e., Eq. (2)] should be adopted.
Figure 7(a) shows that, as expected, the simplified enhancement factor approximates well to the modified enhancement factor when is not too large. But there is an obvious difference between the two when approaches 90°, demonstrating that should be replaced with for a sufficiently large amplitude ratio . For the typical range , the enhancement factor (dashed lines) first increases with . Then it reaches the maximum [22.9–32.9, 1.92–2.76 times that of RPVBGB alone (the field used in Ref. [17])] with a narrow needle width [in the - (-) direction: (), 0.40–0.11 (0.48–0.15) times that of RPVBGB alone] at (corresponding amplitude ratio ) [Figs. 7(b) and 7(c)]. Then it decreases with . At last, it reaches 0 at , equivalent to using APBGB alone.
4. CONCLUSION
In conclusion, an approach is proposed to produce the enhanced superchiral needle by matching the electromagnetic components of the tightly focused combined field formed by superposing an RPVBGB and an APBGB. It is revealed that the ring aperture plays a key role in generating the distribution of the enhancement factor for the superchiral field. The amplitude ratio is an important parameter for adjusting the strength of the superchirality. The phase difference between the two constituent beams makes the superchiral needle rotate. However, the phase difference has no influence on the magnitude of the chirality. This property provides the advantage of being insensitive to the fluctuation of the phase difference, which is important in chirality measurement experiments. Under proper parameters, the enhancement factor can reach from 1.92 to 2.76 times that of the field used in Ref. [17] (i.e., the RPVBGB alone), and the needle width can reach from 0.40 to 0.11 and from 0.48 to 0.15 times that of the field used in Ref. [17] in the - and -directions, respectively. It should be noted that the incident beam is filtered into a narrow annular beam by the ring aperture. Therefore, adjusting the incident waveform (e.g., using a plane wave) would not alter the main conclusions of this study. Since the superchirality is significantly enhanced and the needle width is significantly narrowed, the enhanced superchiral needle would be of interest for the chirality measurement of individual molecules. In addition, we conjecture that it may enable quasi-1D chirality probing and enantioseparation of chiral particles along the beam axis because in the enhanced superchiral needle the longitudinal component is the predominant one.
In addition, the enhanced superchiral needle should be distinguished from the “optimal chiral light” studied in Ref. [31], which focuses on obtaining the optimal optical chirality stemming solely from the longitudinal field components on the beam axis. Therein the optical chirality is represented by the time-averaged helicity density , where is the speed of light in the vacuum. Therefore, the optimization of the optical chirality depends on both a strong electric field and a strong magnetic field in a local region. In Ref. [31], the authors employed a non-vortex radially (azimuthally) polarized beam to generate a longitudinal electric (magnetic) field with a bell-shaped transverse distribution at the beam center, resulting in a strong purely longitudinal electric (magnetic) field on the axis, where the transverse fields vanish. Thus, the strong purely longitudinal electric and magnetic fields together yield the optimal chirality on the axis. Although the “optimal chiral light” obtained in Ref. [31] differs from the enhanced superchiral needle, Ref. [31] demonstrates that purely longitudinal chirality can be generated within a local region of the light field. This reminds us that it might be possible to find purely longitudinal enhanced superchiral needles by strategically matching electromagnetic components from diverse combined fields. It would be meaningful for precise 1D chirality measurement and enantioseparation of chiral particles and would encourage us to do further exploration in the future.
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